Robust Linear Estimation with Covariance Uncertainties

We consider the problem of estimating a random vector x, with covariance uncertainties, that is observed through a known linear transformation H and corrupted by additive noise. We first develop the linear estimator that minimizes the worst-case meansquared error (MSE) across all possible covariance matrices. Although the minimax approach has enjoyed widespread use in the design of robust methods, we show that its performance is often unsatisfactory. We then develop a competitive minimax approach in which we seek the linear estimator that minimizes the worstcase regret, namely, the worst-case difference between the MSE attainable using a linear estimator, ignorant of the signal covariance, and the optimal MSE attained using a linear estimator that knows the signal covariance. We demonstrate, through an example, that the minimax regret approach can improve the performance over the minimax MSE approach.